Ideals in MOD‐ R and the ω‐radical

Let R be an artin algebra, and let mod‐ R denote the category of finitely presented right R ‐modules. The radical rad = rad(mod‐ R ) of this category and its finite powers play a major role in the representation theory of R . The intersection of these finite powers is denoted rad ω , and the nilpotence of this ideal has been investigated, in [ 6, 13 ] for instance. In [ 17 ], arbitrary transfinite powers, rad α , of rad were defined and linked to the extent to which morphisms in mod‐ R may be factorised. In particular, it has been shown that if R is an artin algebra, then the transfinite radical, rad ∞ , the intersection of all ordinal powers of rad, is non‐zero if and only if there is a ‘factorisable system’ of morphisms in rad and, in that case, the Krull–Gabriel dimension of mod‐ R equals ∞ (that is, is undefined). More precise results on the index of nilpotence of rad for artin algebras were proved in [ 14, 20, 24–26 ].